Characters with more Haste do more Damage Per Second because their Class Attacks are faster. For Example, a Shadow Walker with Haste 0% and Sword 1.0 Speed has a Toxic Blade casting time of 1.0s. With Haste 24.03%, the Toxic Blade casting time is 0.76s or +31.63% Damage Per Second.

Since Haste only affects casting time, and most cool downs, you only notice an improvement, when your casting times decreases. Casting times, and cool downs, are rounded up to the nearest .01s. Many classes have Class Attacks with 50% casting time of Basic Attack, but Shadow Walkers have Reflexive Strike, which is 25% casting time of Basic Attack. Reflexive Strike is a good test case to examine the impact on casting time of increasing your Haste Rating. For example when considering a Haste Gem purchase.

Haste Rating has a Rating Cap of Haste 50%, however, a Haste Rating of 1 million, only gives a Haste 49.95% due to diminishing returns.

Note: Haste Stat is the only Stat in the game not affected by Level (Note 1). A Level 1 Shadow Walker, with Haste Rating 925, has the same casting times as a Level 100 Shadow Walker with Haste Rating 925, it is just easier for the Level 100 to get a higher Haste Rating and +31.63% Damage Per Second is much more damage with a Tier 14 Sword than with a Tier 1 Sword.

The following tables contain some of the values for our test case, Reflexive Strike using a Weapon 1.0 Speed. Values for Weapon 1.1- 1.4 Speed will differ in casting time, but not in Bonus Damage Per Second.

Note 1)Haste is unaffected by character Level, unaffected by opponent's level, and has a limit of Haste 50%. It is a Free Standing, Limited Stat Curve or just Limited Stat Curve.Melee and Magic are affected by character Level (Soft Cap), and unaffected by opponent's Level, or Character based Stat Curves.Health is Linear.Remaining Stats are affected by character Level (Soft Cap), and opponent's Level, or Level Based Stat Curves.

Yes, Haste impacts damage over time directly. I have also found that it has a significant effect on overall gameplay when it is complemented with an arsenal of pushback and interrupt skills.

Higher Haste allows the character to seize the first move more often. With a well-timed chain of pushback and interrupt skills, the player can keep the enemy from ever attacking. Lethal hits can be interspersed as the clock allows, with Haste enabling more of these hits to be packed in. Being able to put enemies in a stun lock leads to fewer enemy attacks, less silver spent on potions and healing, and more free silver available for character advancement.